What is the integral of x*sin(sqrt.(x^2 + 4))/sqrt.(x^2 + 4) ?

1 Answer
Jun 19, 2018

#I=-cos(sqrt(x^2+4))+C#

Explanation:

We want to integrate

#I=intx*sin(sqrt(x^2+4))/sqrt(x^2+4)dx#

Make a substitution #color(blue)(u=x^2+4=>du=2xdx#

#I=intx*sin(sqrt(u))/sqrt(u)*1/(2x)du#

#color(white)(I)=1/2intsin(sqrt(u))/sqrt(u)du#

Make a substitution #color(blue)(s=sqrt(u)=>ds=1/(2s)du#

#I=1/2intsin(s)/s*2sds#

#color(white)(I)=intsin(s)ds#

#color(white)(I)=-cos(s)+C#

Substitute back #color(blue)(s=sqrt(u)# and #color(blue)(u=x^2+4#

#I=-cos(sqrt(x^2+4))+C#